Conventionally, receivers use a Hartley-type image rejection mixer or a Weaver-type image rejection mixer for the purpose of removing signal components from the image frequency band.
The receivers in low-end radios have recently come to employ a receiving scheme referred to as “low-IF” which has the ability to lower IF frequency with a single superheterodyne and which readily lends itself to LSI fabrication and signal processing.
Since the IF frequency is low, however, image noise cannot be eliminated by an RF filter inserted in front of the mixer. For this reason, a complex filter circuit has come to be used as the filter circuit for extracting the IF signal from the mixer output signal by employing quadrature demodulation for performing signal processing upon producing a difference in phase of 90° in the signal from the mixer onward.
In particular, wireless LANs and GPS receivers often use a low-IF scheme and there are many cases where such complex filter circuits are used.
In such low-end radios, it has become commonplace that circuit functions, such as from the RF circuit of the transceiver (TRX) to signal processing of the baseband are integrated on a single chip and even the complex filter circuit is now integrated on a chip as a matter of course.
Such a radio was implemented in the form of an MSK-modulation-type digital car radio that used a Gaussian filter in time-division multiplexing that the present inventor and his colleagues were charged with designing. Since then, the digital radio became one that digitizes and transmits even voice. This initial digital radio also came to be referred subsequently as GSM and is the de-facto standard in the most widely produced mobile telephones.
In such a digital radio, inter-symbol interference in the band-limiting filter used in the transceiver has a major effect upon bit error.
After GSM, linear modulation schemes came to be used in addition to GSM. Such a band-limiting filter is implemented by an analog filter and is packaged on a chip.
Accordingly, such a band-limiting filter has become the object, the development of which requires a full-time effort by an LSI designer, and use of Bessel filters for the purpose of data transmission, as in a high-capability mobile phone system in Japan prior to GSM, is no longer seen.
Butterworth filters, elliptic filters or Chebychev filters have come to be used recently, and use of inverse Chebychev filters is seen on very rare occasions.
With a band-limiting filter, it is necessary to hold the amount of attenuation in adjacent bands to 60 or 80 dB and to suppress leakage of signals from adjacent bands to a level on the order of 0.1 or 0.01%.
In the case of a band-limiting filter, therefore, a sharp attenuation characteristic is required and filters such as Bessel filters of the past that have a gentle attenuation characteristic in which importance is attached to the phase characteristic are falling out of use.
In view of an attenuation amount, it is better to select a Butterworth filter over a Bessel filter. If an attempt is made to assure a desired attenuation amount even with a lower order, there is a reason for selecting a Chebychev filter or an elliptic filter over a Butterworth filter.
Although a Butterworth filter generally is said to possess a maximally flat characteristic, if a Butterworth filter and an inverse Chebychev of the same order and having the same cut-off frequency are compared, it will be seen that the inverse Chebychev filter is actually the one that has the maximally flat characteristic.
Similarly, it is well known that if a Chebychev filter and an elliptic filter are compared, it is easier to implement a sharp attenuation characteristic with the elliptic filter.
Further, although an inverse Chebychev filter resembles an elliptic filter, the transfer function thereof is introduced by subjecting the transfer function of a Chebychev filter to an inverse transformation, as will be described later.
In view of the foregoing, an inverse Chebychev filter that maintains a maximally flat characteristic in the passband and affords a sharp attenuation characteristic in the stopband can be said to be the type of filter best suited for use as a band-limiting filter.
However, as will be described below, since the method of deriving the transfer function by inversely transforming the transfer function of a Chebychev filter is itself complicated, the current state of the art is that LSI designers have not yet designed a desirable inverse Chebychev filter as an object of full-time development.
For this reason, inverse Chebychev filters are not as commonplace as Bessel filters, Butterworth filters, Chebychev filter and elliptic filters, whose design techniques are easy to comprehend and which are equipped with tables of numerical data indicating device parameters.
In actuality, the design technique is fairly complicated regardless of whether the filter is the Chebychev filter or elliptical filter, and it is believed difficult for LSI designers to learn these techniques in a short period of time. Nevertheless, since tables indicating device parameters have been available for the last 50 years and characteristics can now be confirmed comparatively easily by simulators and the like, implementation on an LSI chip has at least been accomplished.
A complex filter circuit will be described next. A complex filter circuit is one in which the symmetry between the positive frequency characteristic and the negative frequency characteristic on the frequency axis is eliminated by performing a fixed frequency shift along the frequency axis. This is a filter that includes an imaginary number j. Both the amplitude characteristic and phase characteristic of the filter are represented by a function (of fractional type) of s and are decided by the relationship between the real and imaginary parts obtained by making the substitution s→jω. However, an element having an imaginary part that is not dependent on frequency is introduced and therefore both the amplitude and phase characteristics of the filter change.
In general, an LPF (low pass filter) in the real frequency region is transformed to a BPF (band pass filter) and is used in order to assure an attenuation characteristic in the imaginary frequency region. It is used in a low-IF receiver in which it is necessary to attenuate the image frequency in the receiver.
A complex filter circuit is referred to as a polyphase filter.
If we use H(jω) in the transfer function H(s) of the filter, the amplitude and phase characteristics of the filter are found from the relationship between the real and imaginary parts.
In a complex filter, H(j(ω−ω0)) is used and frequency is shifted by ω0 along the frequency axis to thereby transform an LPF in the real frequency region to a BPF.
A third-order elliptic ladder LPF illustrated in FIG. 26 will be described first.
If we use R1=R3=1 in the transfer function H(s) of a third-order elliptic LC ladder LPF, the transfer function H(s) is found as follows in a case where C2 has been deleted:
                              H          ⁡                      (            s            )                          =                  2                                                    C                1                            ⁢                              L                2                            ⁢                              C                3                            ⁢                              s                3                                      +                                          (                                                      C                    1                                    +                                      C                    3                                                  )                            ⁢                              L                2                            ⁢                              s                2                                      +                                          (                                                      C                    1                                    +                                      L                    2                                    +                                      C                    3                                                  )                            ⁢              s                        +            2                                              (        1        )            
The numerator is set to 2 in such a manner that H(0)=1 will hold.
The transfer function H(s) is as follows in a case where C2 is added on:
                              H          ⁡                      (            s            )                          =                              2            ⁢                          (                                                                    L                    2                                    ⁢                                      C                    2                                    ⁢                                      s                    2                                                  +                1                            )                                                                                                                                                              L                        2                                            ⁡                                              (                                                                                                            C                              1                                                        ⁢                                                          C                              2                                                                                +                                                                                    C                              2                                                        ⁢                                                          C                              3                                                                                +                                                                                    C                              3                                                        ⁢                                                          C                              1                                                                                                      )                                                              ⁢                                          s                      3                                                        +                                                                                                                                                (                                                                        C                          1                                                +                                                  2                          ⁢                                                      C                            2                                                                          +                                                  C                          3                                                                    )                                        ⁢                                          L                      2                                        ⁢                                          s                      2                                                        +                                                            (                                                                        C                          1                                                +                                                  L                          2                                                +                                                  C                          3                                                                    )                                        ⁢                    s                                    +                  2                                                                                        (        2        )            
If the transfer function of Equation (1) is shifted by ω0 along the frequency axis, then we have the following:
                                          H            c                    ⁡                      (            s            )                          =                  2                                                                                                                C                      1                                        ⁢                                          L                      2                                        ⁢                                                                                            C                          3                                                ⁡                                                  (                                                      s                            -                                                          jω                              0                                                                                )                                                                    3                                                        +                                                                                                                                                (                                                                        C                          1                                                +                                                  C                          3                                                                    )                                        ⁢                                                                                            L                          2                                                ⁡                                                  (                                                      s                            -                                                          jω                              0                                                                                )                                                                    2                                                        +                                                            (                                                                        C                          1                                                +                                                                              L                            2                                                    ⁢                                                      C                            3                                                                                              )                                        ⁢                                          (                                              s                        -                                                  jω                          0                                                                    )                                                        +                  2                                                                                        (        3        )            
If the transfer function of Equation (2) is shifted by ω0 along the frequency axis, then we have the following:
                                          H            c                    ⁡                      (            s            )                          =                              2            ⁢                          {                                                                    L                    2                                    ⁢                                                                                    C                        2                                            ⁡                                              (                                                  s                          -                                                      jω                            0                                                                          )                                                              2                                                  +                1                            }                                                                                                                                      C                      1                                        ⁢                                          L                      2                                        ⁢                                                                                            C                          3                                                ⁡                                                  (                                                      s                            -                                                          jω                              0                                                                                )                                                                    3                                                        +                                                            (                                                                        C                          1                                                +                                                  C                          3                                                                    )                                        ⁢                                                                                            L                          2                                                ⁡                                                  (                                                      s                            -                                                          jω                              0                                                                                )                                                                    2                                                        +                                                                                                                                                (                                                                        C                          1                                                +                                                  L                          2                                                +                                                  C                          3                                                                    )                                        ⁢                                          (                                              s                        -                                                  jω                          0                                                                    )                                                        +                  2                                                                                        (        4        )            
That, we have the following:
                                          H            c                    ⁡                      (            jω            )                          =                              H            ⁢                          (                        ⁢                          j              ⁡                              (                                  ω                  -                                      ω                    0                                                  )                                              =                                    2              ⁢                              {                                                                            L                      2                                        ⁢                                                                                            C                          2                                                ⁡                                                  (                                                      j                            ⁡                                                          (                                                              ω                                -                                                                  ω                                  0                                                                                            )                                                                                )                                                                    2                                                        +                  1                                }                                                                                                                                                        C                        1                                            ⁢                                              L                        2                                            ⁢                                                                                                    C                            3                                                    ⁡                                                      (                                                          j                              ⁡                                                              (                                                                  ω                                  -                                                                      ω                                    0                                                                                                  )                                                                                      )                                                                          3                                                              +                                                                  (                                                                              C                            1                                                    +                                                      C                            3                                                                          )                                            ⁢                                                                                                    L                            2                                                    ⁡                                                      (                                                          j                              ⁡                                                              (                                                                  ω                                  -                                                                      ω                                    0                                                                                                  )                                                                                      )                                                                          2                                                              +                                                                                                                                                                  (                                                                              C                            1                                                    +                                                      L                            2                                                    +                                                      C                            3                                                                          )                                            ⁢                                              (                                                  j                          ⁡                                                      (                                                          ω                              -                                                              ω                                0                                                                                      )                                                                          )                                                              +                    2                                                                                                          (        5        )            
A complex elliptic filter having the transfer function of Equation (5) is illustrated in FIG. 4. Here G1=ω0C1, G2=ω0C2, G3=ω0C3, R2=ω0L2 holds, wherein G1, G2 and G3 and R2 are assumed to be positive (>0).
By way of example, if a third-order elliptic LPF (C1=C3=0.6229, C2=0.0240, L2=0.9477, R1=R3=1, ωc=1.823 rad) is shifted along the frequency axis by ω0=1.92 rad, the filter will be transformed to a BPF having a passband of 0.097 to 3.743 rad and a center frequency of ω0=1.92 rad, as illustrated in FIG. 27. This is a so-called complex BPF having an asymmetric characteristic with respect to the ω=0 frequency axis.
Thus, the frequency characteristic or phase characteristic of a complex BPF is obtained in simple fashion if a software tool such as Excel, is used to perform a frequency shift of ω→ω−ω0.
However, as shown in FIG. 4, the filter structure is complicated. Moreover, a negative conductance and a negative resistance are required, and implementation using passive elements is not possible.
The circuit diagram of a third-order gm-C LPF that emulates an RLC ladder filter is illustrated in FIG. 28. If the elliptic capacitance C2 is deleted, the transfer function H(s) becomes as follows in the case of a single-ended arrangement:
                              H          ⁡                      (            s            )                          =                                            g                              m                ⁢                                                                  ⁢                1                                      ⁢                          g                              m                ⁢                                                                  ⁢                3                                      ⁢                          g                              m                ⁢                                                                  ⁢                6                                                                                                                                                    C                      1                                        ⁢                                          C                                              L                        ⁢                                                                                                  ⁢                        2                                                              ⁢                                          C                      3                                        ⁢                                          s                      3                                                        +                                                                                    C                                                  L                          ⁢                                                                                                          ⁢                          2                                                                    ⁡                                              (                                                                                                            g                                                              m                                ⁢                                                                                                                                  ⁢                                2                                                                                      ⁢                                                          C                              3                                                                                +                                                                                    g                                                              m                                ⁢                                                                                                                                  ⁢                                7                                                                                      ⁢                                                          C                              1                                                                                                      )                                                              ⁢                                          s                      2                                                        +                                                                                                                                                (                                                                                                    g                                                          m                              ⁢                                                                                                                          ⁢                              5                                                                                ⁢                                                      g                                                          m                              ⁢                                                                                                                          ⁢                              6                                                                                ⁢                                                      C                            1                                                                          +                                                                              g                                                          m                              ⁢                                                                                                                          ⁢                              2                                                                                ⁢                                                      g                                                          m                              ⁢                                                                                                                          ⁢                              7                                                                                ⁢                                                      C                                                          L                              ⁢                                                                                                                          ⁢                              2                                                                                                      +                                                                              g                                                          m                              ⁢                                                                                                                          ⁢                              3                                                                                ⁢                                                      g                                                          m                              ⁢                                                                                                                          ⁢                              4                                                                                ⁢                                                      C                            3                                                                                              )                                        ⁢                    s                                    +                                                                                                      (                                                                                    g                                                  m                          ⁢                                                                                                          ⁢                          2                                                                    ⁢                                              g                                                  m                          ⁢                                                                                                          ⁢                          5                                                                    ⁢                                              g                                                  m                          ⁢                                                                                                          ⁢                          6                                                                                      +                                                                  g                                                  m                          ⁢                                                                                                          ⁢                          3                                                                    ⁢                                              g                                                  m                          ⁢                                                                                                          ⁢                          4                                                                    ⁢                                              g                                                  m                          ⁢                                                                                                          ⁢                          7                                                                                                      )                                                                                        (        6        )            Here a correction for setting H(0)=1 has not been applied.
In the case of a double-ended arrangement (fully differential), we have the following:
                              H          ⁡                      (            s            )                          =                              8            ⁢                          g                              m                ⁢                                                                  ⁢                1                                      ⁢                          g                              m                ⁢                                                                  ⁢                3                                      ⁢                          g                              m                ⁢                                                                  ⁢                6                                                                                                                                                    C                      1                                        ⁢                                          C                                              L                        ⁢                                                                                                  ⁢                        2                                                              ⁢                                          C                      3                                        ⁢                                          s                      3                                                        +                                      2                    ⁢                                                                  C                                                  L                          ⁢                                                                                                          ⁢                          2                                                                    ⁡                                              (                                                                                                            g                                                              m                                ⁢                                                                                                                                  ⁢                                2                                                                                      ⁢                                                          C                              3                                                                                +                                                                                    g                                                              m                                ⁢                                                                                                                                  ⁢                                7                                                                                      ⁢                                                          C                              1                                                                                                      )                                                              ⁢                                          s                      2                                                        +                                                                                                                          4                    ⁢                                          (                                                                                                    g                                                          m                              ⁢                                                                                                                          ⁢                              3                                                                                ⁢                                                      g                                                          m                              ⁢                                                                                                                          ⁢                              6                                                                                ⁢                                                      C                            1                                                                          +                                                                              g                                                          m                              ⁢                                                                                                                          ⁢                              2                                                                                ⁢                                                      g                                                          m                              ⁢                                                                                                                          ⁢                              7                                                                                ⁢                                                      C                                                          L                              ⁢                                                                                                                          ⁢                              2                                                                                                      +                                                                              g                                                          m                              ⁢                                                                                                                          ⁢                              3                                                                                ⁢                                                      g                                                          m                              ⁢                                                                                                                          ⁢                              4                                                                                ⁢                                                      C                            3                                                                                              )                                        ⁢                    s                                    +                                                                                                      8                  ⁢                                      (                                                                                            g                                                      m                            ⁢                                                                                                                  ⁢                            2                                                                          ⁢                                                  g                                                      m                            ⁢                                                                                                                  ⁢                            5                                                                          ⁢                                                  g                                                      m                            ⁢                                                                                                                  ⁢                            6                                                                                              +                                                                        g                                                      m                            ⁢                                                                                                                  ⁢                            3                                                                          ⁢                                                  g                                                      m                            ⁢                                                                                                                  ⁢                            4                                                                          ⁢                                                  g                                                      m                            ⁢                                                                                                                  ⁢                            7                                                                                                                )                                                                                                          (        7        )            
If gm1 to gm7 are equal and denoted by gm, then H(s) will be expressed as follows in case of the single-ended arrangement:
                              H          ⁡                      (            s            )                          =                  1                                                                                          C                    1                                    ⁢                                      C                                          L                      ⁢                                                                                          ⁢                      2                                                        ⁢                                      C                    3                                                                    g                  m                  3                                            ⁢                              s                3                                      +                                                            C                                      L                    ⁢                                                                                  ⁢                    2                                                                    g                  m                  2                                            ⁢                              (                                                      C                    1                                    +                                      C                    3                                                  )                            ⁢                              s                2                                      +                                          (                                                                            C                      1                                                              g                      m                                                        +                                                            C                                              L                        ⁢                                                                                                  ⁢                        2                                                                                    g                      m                                                        +                                                            C                      3                                                              g                      m                                                                      )                            ⁢              s                        +            2                                              (        8        )            
If gm1 to gm7 are equal and denoted by gm, then H(s) will be expressed as follows in case of the double-ended (fully differential) arrangement:
                              H          ⁡                      (            s            )                          =                  1                                                                                          C                    1                                    ⁢                                      C                                          L                      ⁢                                                                                          ⁢                      2                                                        ⁢                                      C                    3                                                                    8                  ⁢                                      g                    m                    3                                                              ⁢                              s                3                                      +                                                            C                                      L                    ⁢                                                                                  ⁢                    2                                                                    4                  ⁢                                      g                    m                    2                                                              ⁢                              (                                                      C                    1                                    +                                      C                    3                                                  )                            ⁢                              s                2                                      +                                          (                                                                            C                      1                                                              2                      ⁢                                              g                        m                                                                              +                                                            C                                              L                        ⁢                                                                                                  ⁢                        2                                                                                    2                      ⁢                                              g                        m                                                                              +                                                            C                      3                                                              2                      ⁢                                              g                        m                                                                                            )                            ⁢              s                        +            2                                              (        9        )            
In the case of an elliptic gm-C LPF onto which the elliptic capacitor C2 has been added, the transfer function H(s) is as follows in the case of the single-ended arrangement:
                              H          ⁡                      (            s            )                          =                                            g                              m                ⁢                                                                  ⁢                1                                      ⁢                                          g                                  m                  ⁢                                                                          ⁢                  6                                            ⁡                              (                                                                            C                      2                                        ⁢                                          C                                              L                        ⁢                                                                                                  ⁢                        2                                                              ⁢                                          s                      2                                                        +                                                            g                                              m                        ⁢                                                                                                  ⁢                        3                                                              ⁢                                          g                                              m                        ⁢                                                                                                  ⁢                        6                                                                                            )                                                                                                                              C                                          L                      ⁢                                                                                          ⁢                      2                                                        (                                                                                    g                                                  m                          ⁢                                                                                                          ⁢                          6                                                                    ⁢                                              C                        1                                            ⁢                                              C                        2                                                              +                                                                  g                                                  m                          ⁢                                                                                                          ⁢                          6                                                                    ⁢                                              C                        1                                            ⁢                                              C                        3                                                              -                                                                                                                                                                                                                                                g                                                          m                              ⁢                                                                                                                          ⁢                              6                                                                                ⁢                                                      C                            2                                                    ⁢                                                      C                            3                                                                          +                                                  2                          ⁢                                                      g                                                          m                              ⁢                                                                                                                          ⁢                              4                                                                                ⁢                                                      C                            2                                                    ⁢                                                      C                            3                                                                          +                                                  2                          ⁢                                                      g                                                          m                              ⁢                                                                                                                          ⁢                              4                                                                                ⁢                                                      C                            2                            2                                                    ⁢                                                      C                                                          L                              ⁢                                                                                                                          ⁢                              2                                                                                                                          )                                        ⁢                                          s                      3                                                        +                                                                                                                          C                                          L                      ⁢                                                                                          ⁢                      2                                                        (                                                                                    g                                                  m                          ⁢                                                                                                          ⁢                          2                                                                    ⁢                                              g                                                  m                          ⁢                                                                                                          ⁢                          6                                                                    ⁢                                              C                        3                                                              +                                                                  g                                                  m                          ⁢                                                                                                          ⁢                          2                                                                    ⁢                                              g                                                  m                          ⁢                                                                                                          ⁢                          6                                                                    ⁢                                              C                        2                                                              +                                                                                                                                                                                                                                                g                                                          m                              ⁢                                                                                                                          ⁢                              6                                                                                ⁢                                                      g                                                          m                              ⁢                                                                                                                          ⁢                              7                                                                                ⁢                                                      C                            1                                                                          -                                                                              g                                                          m                              ⁢                                                                                                                          ⁢                              6                                                                                ⁢                                                      g                                                          m                              ⁢                                                                                                                          ⁢                              7                                                                                ⁢                                                      C                            2                                                                          +                                                  2                          ⁢                                                      g                                                          m                              ⁢                                                                                                                          ⁢                              4                                                                                ⁢                                                      g                                                          m                              ⁢                                                                                                                          ⁢                              7                                                                                ⁢                                                      C                            2                                                                                              )                                        ⁢                                          s                      2                                                        +                                                                                                                          g                                          m                      ⁢                                                                                          ⁢                      6                                                        (                                                                                    g                                                  m                          ⁢                                                                                                          ⁢                          5                                                                    ⁢                                              g                                                  m                          ⁢                                                                                                          ⁢                          6                                                                    ⁢                                              C                        1                                                              +                                                                  g                                                  m                          ⁢                                                                                                          ⁢                          2                                                                    ⁢                                              g                                                  m                          ⁢                                                                                                          ⁢                          7                                                                    ⁢                                              C                                                  L                          ⁢                                                                                                          ⁢                          2                                                                                      +                                                                  g                                                  m                          ⁢                                                                                                          ⁢                          3                                                                    ⁢                                              g                                                  m                          ⁢                                                                                                          ⁢                          4                                                                    ⁢                                              C                        3                                                              +                                                                  g                                                  m                          ⁢                                                                                                          ⁢                          3                                                                    ⁢                                              g                                                  m                          ⁢                                                                                                          ⁢                          6                                                                    ⁢                                              C                        2                                                              +                                                                                                                                                                                                                                                g                                                          m                              ⁢                                                                                                                          ⁢                              3                                                                                ⁢                                                      g                                                          m                              ⁢                                                                                                                          ⁢                              4                                                                                ⁢                                                      C                            2                                                                          +                                                                              g                                                          m                              ⁢                                                                                                                          ⁢                              4                                                                                ⁢                                                      g                                                          m                              ⁢                                                                                                                          ⁢                              5                                                                                ⁢                                                      C                            2                                                                          -                                                                              g                                                          m                              ⁢                                                                                                                          ⁢                              5                                                                                ⁢                                                      g                                                          m                              ⁢                                                                                                                          ⁢                              6                                                                                ⁢                                                      C                            2                                                                                              )                                        ⁢                    s                                    +                                                                                                                          g                                          m                      ⁢                                                                                          ⁢                      6                                                        ⁡                                      (                                                                                            g                                                      m                            ⁢                                                                                                                  ⁢                            2                                                                          ⁢                                                  g                                                      m                            ⁢                                                                                                                  ⁢                            5                                                                          ⁢                                                  g                                                      m                            ⁢                                                                                                                  ⁢                            6                                                                                              +                                                                        g                                                      m                            ⁢                                                                                                                  ⁢                            3                                                                          ⁢                                                  g                                                      m                            ⁢                                                                                                                  ⁢                            4                                                                          ⁢                                                  g                                                      m                            ⁢                                                                                                                  ⁢                            7                                                                                                                )                                                                                                          (        10        )            Here a correction for setting H(0)=1 has not been applied.
In the case of the double-ended (fully differential) arrangement, we have the following:
                              H          ⁡                      (            s            )                          =                              2            ⁢                          g                              m                ⁢                                                                  ⁢                1                                      ⁢                                          g                                  m                  ⁢                                                                          ⁢                  6                                            ⁡                              (                                                                            C                      2                                        ⁢                                          C                                              L                        ⁢                                                                                                  ⁢                        2                                                              ⁢                                          s                      2                                                        +                                      4                    ⁢                                          g                                              m                        ⁢                                                                                                  ⁢                        3                                                              ⁢                                          g                                              m                        ⁢                                                                                                  ⁢                        6                                                                                            )                                                                                                                              C                                          L                      ⁢                                                                                          ⁢                      2                                                        (                                                                                    g                                                  m                          ⁢                                                                                                          ⁢                          6                                                                    ⁢                                              C                        1                                            ⁢                                              C                        2                                                              +                                                                  g                                                  m                          ⁢                                                                                                          ⁢                          6                                                                    ⁢                                              C                        1                                            ⁢                                              C                        3                                                              -                                                                                                                                                                                                                                                g                                                          m                              ⁢                                                                                                                          ⁢                              6                                                                                ⁢                                                      C                            2                                                    ⁢                                                      C                            3                                                                          +                                                  2                          ⁢                                                      g                                                          m                              ⁢                                                                                                                          ⁢                              4                                                                                ⁢                                                      C                            2                                                    ⁢                                                      C                            3                                                                          +                                                  2                          ⁢                                                      g                                                          m                              ⁢                                                                                                                          ⁢                              4                                                                                ⁢                                                      C                            2                            2                                                    ⁢                                                      C                                                          L                              ⁢                                                                                                                          ⁢                              2                                                                                                                          )                                        ⁢                                          s                      3                                                        +                                                                                                      2                  ⁢                                                            C                                              L                        ⁢                                                                                                  ⁢                        2                                                              (                                                                                            g                                                      m                            ⁢                                                                                                                  ⁢                            2                                                                          ⁢                                                  g                                                      m                            ⁢                                                                                                                  ⁢                            6                                                                          ⁢                                                  C                          3                                                                    +                                                                        g                                                      m                            ⁢                                                                                                                  ⁢                            2                                                                          ⁢                                                  g                                                      m                            ⁢                                                                                                                  ⁢                            6                                                                          ⁢                                                  C                          2                                                                    +                                                                                                                                                                                                                                                                    g                                                          m                              ⁢                                                                                                                          ⁢                              6                                                                                ⁢                                                      g                                                          m                              ⁢                                                                                                                          ⁢                              7                                                                                ⁢                                                      C                            1                                                                          -                                                                              g                                                          m                              ⁢                                                                                                                          ⁢                              6                                                                                ⁢                                                      g                                                          m                              ⁢                                                                                                                          ⁢                              7                                                                                ⁢                                                      C                            2                                                                          +                                                  2                          ⁢                                                      g                                                          m                              ⁢                                                                                                                          ⁢                              4                                                                                ⁢                                                      g                                                          m                              ⁢                                                                                                                          ⁢                              7                                                                                ⁢                                                      C                            2                                                                                              )                                        ⁢                                          s                      2                                                        +                                                                                                      4                  ⁢                                                            g                                              m                        ⁢                                                                                                  ⁢                        6                                                              (                                                                                            g                                                      m                            ⁢                                                                                                                  ⁢                            5                                                                          ⁢                                                  g                                                      m                            ⁢                                                                                                                  ⁢                            6                                                                          ⁢                                                  C                          1                                                                    +                                                                        g                                                      m                            ⁢                                                                                                                  ⁢                            2                                                                          ⁢                                                  g                                                      m                            ⁢                                                                                                                  ⁢                            7                                                                          ⁢                                                  C                                                      L                            ⁢                                                                                                                  ⁢                            2                                                                                              +                                                                        g                                                      m                            ⁢                                                                                                                  ⁢                            3                                                                          ⁢                                                  g                                                      m                            ⁢                                                                                                                  ⁢                            4                                                                          ⁢                                                  C                          3                                                                    +                                                                        g                                                      m                            ⁢                                                                                                                  ⁢                            3                                                                          ⁢                                                  g                                                      m                            ⁢                                                                                                                  ⁢                            6                                                                          ⁢                                                  C                          2                                                                    +                                                                                                                                                                                                                                                                    g                                                          m                              ⁢                                                                                                                          ⁢                              3                                                                                ⁢                                                      g                                                          m                              ⁢                                                                                                                          ⁢                              4                                                                                ⁢                                                      C                            2                                                                          +                                                                              g                                                          m                              ⁢                                                                                                                          ⁢                              4                                                                                ⁢                                                      g                                                          m                              ⁢                                                                                                                          ⁢                              5                                                                                ⁢                                                      C                            2                                                                          -                                                                              g                                                          m                              ⁢                                                                                                                          ⁢                              5                                                                                ⁢                                                      g                                                          m                              ⁢                                                                                                                          ⁢                              6                                                                                ⁢                                                      C                            1                                                                                              )                                        ⁢                    s                                    +                                                                                                      8                  ⁢                                                            g                                              m                        ⁢                                                                                                  ⁢                        6                                                              ⁡                                          (                                                                                                    g                                                          m                              ⁢                                                                                                                          ⁢                              2                                                                                ⁢                                                      g                                                          m                              ⁢                                                                                                                          ⁢                              5                                                                                ⁢                                                      g                                                          m                              ⁢                                                                                                                          ⁢                              6                                                                                                      +                                                                              g                                                          m                              ⁢                                                                                                                          ⁢                              3                                                                                ⁢                                                      g                                                          m                              ⁢                                                                                                                          ⁢                              4                                                                                ⁢                                                      g                                                          m                              ⁢                                                                                                                          ⁢                              7                                                                                                                          )                                                                                                                              (        11        )            
If gm1 to gm7 are equal and denoted by gm, then H(s) will be expressed as follows in case of the single-ended arrangement:
                              H          ⁡                      (            s            )                          =                                                                                                  C                    2                                    ⁢                                      C                                          L                      ⁢                                                                                          ⁢                      2                                                                                        g                  m                  2                                            ⁢                              s                2                                      +            1                                                                                                                                                  C                                                  L                          ⁢                                                                                                          ⁢                          2                                                                                            g                        m                        3                                                              ⁢                                          (                                                                                                    C                            1                                                    ⁢                                                      C                            2                                                                          +                                                                              C                            1                                                    ⁢                                                      C                            3                                                                          +                                                                              C                            2                                                    ⁢                                                      C                            3                                                                          +                                                  2                          ⁢                                                      C                            2                            2                                                                                              )                                        ⁢                                          s                      3                                                        +                                                                                                                                                                        C                                                  L                          ⁢                                                                                                          ⁢                          2                                                                                            g                        m                        2                                                              ⁢                                          (                                                                        C                          1                                                +                                                  2                          ⁢                                                      C                            2                                                                          +                                                  C                          3                                                                    )                                        ⁢                                          s                      2                                                        +                                                            (                                                                                                    C                            1                                                                                g                            m                                                                          +                                                                              C                                                          L                              ⁢                                                                                                                          ⁢                              2                                                                                                            g                            m                                                                          +                                                                              C                            3                                                                                g                            m                                                                          +                                                                              2                            ⁢                                                          C                              2                                                                                                            g                            m                                                                                              )                                        ⁢                    s                                    +                  2                                                                                        (        12        )            
If gm1 to gm7 are equal and denoted by gm, then H(s) will be expressed as follows in case of the double-ended (fully differential) arrangement:
                              H          ⁡                      (            s            )                          =                                                                                                  C                    2                                    ⁢                                      C                                          L                      ⁢                                                                                          ⁢                      2                                                                                        4                  ⁢                                      g                    m                    2                                                              ⁢                              s                2                                      +            1                                                                                                                                                  C                                                  L                          ⁢                                                                                                          ⁢                          2                                                                                            8                        ⁢                                                  g                          m                          3                                                                                      ⁢                                          (                                                                                                    C                            1                                                    ⁢                                                      C                            2                                                                          +                                                                              C                            1                                                    ⁢                                                      C                            3                                                                          +                                                                              C                            2                                                    ⁢                                                      C                            3                                                                          +                                                  2                          ⁢                                                      C                            2                            2                                                                                              )                                        ⁢                                          s                      3                                                        +                                                                                                                                                                        C                                                  L                          ⁢                                                                                                          ⁢                          2                                                                                            4                        ⁢                                                  g                          m                          2                                                                                      ⁢                                          (                                                                        C                          1                                                +                                                  2                          ⁢                                                      C                            2                                                                          +                                                  C                          3                                                                    )                                        ⁢                                          s                      2                                                        +                                                                                                                                                (                                                                                                    C                            1                                                                                2                            ⁢                                                          g                              m                                                                                                      +                                                                              C                                                          L                              ⁢                                                                                                                          ⁢                              2                                                                                                            2                            ⁢                                                          g                              m                                                                                                      +                                                                              C                            3                                                                                2                            ⁢                                                          g                              m                                                                                                      +                                                                              C                            2                                                                                g                            m                                                                                              )                                        ⁢                    s                                    +                  2                                                                                        (        13        )            
In a gm-C filter, an inductance L is converted to a capacitance C by a gyrator and therefore the element that contributes to frequency is a capacitance only. Accordingly, a complex filter can be constructed by adding on only a negative conductance and a negative resistance is unnecessary.
A capacitance C possesses a conductance element G that is dependent upon frequency ω as is expressed as follows:G=jωC  (14)
If we writeG(s)=sC  (15)then we have the following:G(jω)=jωC  (16)
Accordingly, a complex conductance G(j(ω−ω0)) of the capacitance C that constructs the complex filter becomes as follows:G(j(ω−ω0))=j(ω−ω0)C=jωC−jω0C  (17)
In a quadrature demodulation system, filters for the two channels I and Q are required. Between the I-channel filter and the Q-channel filter, the input signals have a phase difference of 90°. If the input signal to the I-channel filter is V, then the input signal to the Q-channel filter is jV.
Further, in two identical filters, the voltages applied to the respective capacitances have a phase difference of 90° between them and are represented by Vi and jVi.
That is, it is written [see FIGS. 29A and 29B] that the complex conductance of Equation (17) can be implemented by mutually coupling the conductances of the I and Q channels by two OTAs (Operational Transconductance Amplifiers) for which gm=−ω0C holds.
The filter is as illustrated at FIG. 29C in actuality. Alternatively, FIG. 4 is illustrated also by an equivalent circuit of the kind shown in FIG. 31 using a gyrator.
In comparison with FIG. 4, however, the only difference is the absence of −jGi (=−jω0Ci) connected in parallel with a capacitance Ci and the current −jGi that is supplied from an OTA. At a glance, it may appear that this is at variance with Kirchhoff's Law because there is no current path.
However, the current of −jGi (=−jω0Ci) is negative and there is no flow of current in −jGi of FIG. 4. There is equivalence because current flows out of −jGi.
Further, in a gm-C LPF, the OTA is tuned in such a manner that the ratio gm/Ci between gm and capacitance Ci of the OTA will be constant. However, tuning is performed in such a manner that the ratio gm/Ci between gm (=−ω0Ci) of the OTA, which couples the capacitances, and capacitance Ci will be constant (−ω0).
Examples of a complex gm-C BPF using the transformation shown in FIG. 29C is nowhere to be found if limited to the case of the elliptic filter.
FIG. 30 illustrates a case where an elliptic capacitor C2 it taken up. In FIG. 30, gmC2=ω0CL2 holds. Further, since C2<<C1, C3 holds with regard to the elliptic capacitor C2, G2<<G1, G3 holds in FIG. 4.
Furthermore, in order to reduce the number of OTAs, implementing a complex elliptic gm-C filter by a biquad arrangement of the kind shown in FIG. 32 is also conceivable.
Although it will suffice to couple the elliptic capacitors C2 using a differential OTA, here it is assumed that a load will be placed upon gmc1 and gmc3.
Even if G2 is doubled andgm1=ω0(C1+C2),gm3=ω0(C3+C2)hold, it can be expected that there will be no major shift in the relative position of the transmission zero point.
However, since multiplication by 2 is performed in the term containing C22 (C2<<CL2) in the coefficients of s3 in the denominator of Equation (12), some error is produced. (A biquad elliptic gm-C LPF will be described in detail later.)
The foregoing is a method of constructing an ordinary complex filter by a gm-C filter.
In the paper by F. Behbahani et al. cited as an example of the prior art, the above-described complex third-order elliptic gm-C LPF is raised to one of the fifth order. The circuit thereof is shown in FIG. 33.
Similarly, in this example, a coupler that couples the elliptic capacitors between the I and Q channels is not illustrated intentionally. The reason may be that since the circuit is single-ended and the OTA is not displayed as having a differential input, the coupler between the elliptic capacitors requiring a differential OTA cannot be displayed either.
Although Non-Patent Document 2 describes the methods of Voorman and Shouno, the content is completely different from that of the present invention described later.
[Non-Patent Document 1]
F. Behbahani et al., “A Fully Integrated Low-IF CMOS GPS Radio With On-Chip Analog Image Rejection,” IEEE Journal of Solid-State Circuits, Vol. 37, No. 12, pp. 1721-1727. December 2002. 2000/7/26-27, pp. 222-224, FIG. 1.
[Non-Patent Document 2]
Hiroshi Tanimoto, “Precautions Relating to Equivalency of the Voorman Method and Shouno Method in Implementation of Complex Filters”, 00/10/21 Hokkaido Branch Meeting, Kitami Institute of Technology, Internet Site: <URL:http://www.elec.kitami13it.ac.jp/{tilde over ( )}tanimoto/PUBLICATIONS/comp lex_filter95.pdf>